Math tools and strategies for today's standards…that lay the foundation for tomorrow!

Month: January 2016

When I reflect on one thing that I think has the potential to change the math journey for every child, I think of Number Talks! I first learned about Number Talks when I purchased Sherry Parrish’s Number Talks book. I loved that it included example dot patterns, 10frame patterns, rekenrek patterns, and examples for every operation that I could use in my classroom as well as a DVD showing students actually doing number talks from kindergarten to 5th grade. I was floored with their thinking! There were some times I had to pause the video to try to figure out what the children were doing to solve some pretty sophisticated expressions! Wow!

I also love the book Making Number Talks Matter by Cathy Humphreys and Ruth Parker because it not only discusses the strategies that students employ but it also shows how the very same strategies used for simple expressions are applied down the road to larger numbers, decimals, and fractions! So incredibly powerful!!

Here are the reasons why I love Number Talks so much:

students are actively constructing their own thinking in a way that makes sense to them

number talk routines get children talking about numbers and their thinking – so many times I have learned about a way of solving a problem that I never would have thought of on my own!

students learn that there are many ways to solve a problem (not just one as those who are taught with the algorithm come to believe) and respect the perspectives of others

students learn from each other and even learn to critique the work of others as they explain their strategies

working on thinking strategically through number talks unlocks the ability to solve problems children would never have previously thought they could! I have asked children who have memorized their math facts to solve 4 x 12 and they have told me, “I haven’t learned that one yet.” When I ask students who have had number talks to solve 4 x 12, they immediately begin thinking about how they can figure it out…and they have a couple of ways of doing it!

students are given the opportunity to make connections between topics that would never have happened when focusing only on an algorithmic way of solving problems….one 4th grade classroom I work in was working on the angles within a circle. When we did a number talk for 45 x 16, one student answered 720 pretty quickly. He explained that since he knew there were 8-45 degree angles in a circle, 16 would be 2 circles worth of degrees. Amazing!

number talks allow me to talk to students about having a growth mindset (I love Carol Dweck and Jo Boaler’s work on this!)….students know that if they make a mistake when trying out a new strategy they are growing their brains!

students can enter the activity at their readiness level – if I give 7 + 8, students who are still counting on can solve it…but it also allows students who realize a doubles plus/minus 1 strategy could be used or even students who take some away from the other number to make a 10 are all successful and respected!

When most people are asked how to define fluency, they think of speed and accuracy. I think everyone would agree that students who have automatized their math facts are in a much better position to learn more difficult concepts because they have more brain energy to concentrate on understanding the concept they are learning rather than worrying about the math facts embedded in the problem. There is a third part to this definition, though, that is the key to laying the foundation for future math learning – flexibility! When children are taught strategies that allow them think flexibly about numbers, we are helping them develop their number sense and unlocking students’ access to more difficult concepts later. So not only are we helping them learn the math facts, but those very same strategies can be applied to larger numbers, and even fractions and decimals down the road.

In our district, we have been following a progression of strategies for each operation developed by Dr. Nicki Newton in her upcoming book Running Records due out in the spring (2016). I took a webinar this past summer to learn about this instructional assessment tool and it has simply transformed everything! Combined with implementing Number Talks, I believe we have the chance to change the trajectory of math education for our students.

Essentially, the running record is a math interview where we get a measure on the students’ accuracy, flexibility using strategies, and speed (although this part is not nearly as important to me as accuracy and flexibility since I believe, like Jo Boaler’s, that you don’t need to be fast at math to be good at math). Once we figure out where the student falls on the progression, I have been developing a collection of games and activities for each strategy so we can begin working with the student on the strategy they need and help them move forward on the progression. It has been awesome to watch the progress our students are making!

To teach the strategies, we use math tools such as 10 frames, Cuisenaire rods, rekenreks, counting objects and dry erase board 10 frames. Students need to see concretely how and why this strategies work. I have created a set of videos (click here for the webpage) that shows exactly how we are using these tools to teach these strategies. Here is the progression of the strategies we are using:

Plus 1 – We want student to understand that adding one to a number results in the next counting number.

Plus 0 – We tend to combine the Plus 0 examples with the Plus 1 examples since adding 0 is such a difficult concept for our young kiddos.

Count on 2 or 3 – For this strategy, we focus on starting with the larger number especially when the larger number comes second…so 2 + 6 I would want my student to start at the 6 and count on 2. The last thing we want to do is encourage children to count, but if we are just adding 2 or 3 and we have this strategy in place, it will take about the 3 seconds we are looking for to get an accurate answer.

Adding within 5 – We want to be sure that students have a good handle on adding numbers whose sums are up to 5 before we move on the sums within 10 so this is a great checking point.

Adding within 10 – We want to be sure students are comfortable working with sums within 10 before moving on to the next strategy.

Adding to make 10 – The most crucial skill! You can’t practice it often enough even when children pass this strategy. This is a strategy in which we want automaticity. Our students play lots of games like “Go Ten” which is played just like Go Fish but you are asking your partner for the pair that makes 10 with yours or we place 12 cards face up on a table and students are choosing two that combine to make 10. We assess this strategy by giving the students a number and they tell us the number that adds with it to make 10. Before we move a student on from this strategy, we want to be sure that if we say “2” they say “8” and so on without hesitating. This forms such an important foundation for future math concepts.

(disclaimer here – at this point with our K-2 kiddos, we then start them on the subtraction strategies within 10…once they finish those, they reenter this progression here)

Plus 10 – we want our student to develop automaticity with 10 plus any single-digit number since that, too, is a foundational skill embedded in more difficult concepts later on

Doubles – the last of the strategies that we look for automaticity since it is also a foundational skill for future strategies

Doubles plus 1 – We want students to recognize that if they are adding two numbers that are 1 apart, they can double one of the numbers and adjust accordingly. For example, 6 + 7 can be thought of as 6 + 6 + 1 or some children prefer to think of it as 7 + 7 – 1.

Doubles plus 2 – We want students to recognize that if they are adding two numbers that are two apart, they can do Doubles Plus 2, Doubles Minus 2 or even Double the middle. Let’s say we have 6 + 8….Student 1 can think 6 + 6 + 2 = 14, Student 2 could think 8 + 8 – 2 = 14, and Student 3 could think of taking one away from the 8 and giving it to the 6 to change the expression to 7 + 7 = 14. There is no one right strategy..the idea is that children are thinking flexibility about numbers and are using a strategy easiest for them.

Decompose to make 10 when adding a 9 – These last two strategies are the most important strategies because they translate perfectly to larger numbers, adding fractions and decimals, and within concepts of adding units of measurement. Since one of the addends is a 9, we can take away 1 from the other addend and rename the expression with a 10 and whatever is left over. For example, 9 + 6 would be changed into 10 + 5 to make it easier for our brains. Focusing on just the 9 first helps the students develop the ability to think about taking an amount from the other number to make a 10.

Decompose to make 10 when adding a 7 or 8 – This is the last strategy on the progression and forms the foundation for our work with larger numbers down the line. We need to take either 3 from the other number to make a 10 with the 7 or 2 from the other number to make a 10 with the 8.

The decomposing strategy with 7, 8 or 9 is truly the most versatile, so if a child is using this strategy when they are working with any of the ones that fit into doubles plus 1 or doubles plus 2 I would not make students work on the doubles strategies. But, I have found students to stick with the doubles strategy for many expressions such as 5 + 8. Students will say they know 5 + 5 + 3 makes 13. For them, I do want to work with them on decomposing to make a 10 since it is so versatile to future math concepts.

One of the things I feel passionate about is teaching math facts strategically even with our youngest students. Not only are we teaching them strategies for figuring out the particular math facts that are appropriate for their development, but there are two additional bonuses: we are teaching about the conceptual meaning of the operation itself AND we are providing our students with the confidence to try to solve problems they never thought they could (as opposed to students who have learned by memorizing math facts and won’t even attempt more difficult ones because they say they “haven’t learned that one yet”).

Recently I worked with some students to understand the strategy of double, double, double with their multiplication math facts of 8. One of my favorite and, I believe, most powerful math tools is Cuisenaire rods because they help children see that numbers exist as a group rather than a collection ones that need to be counted. I particularly love them to show multiplication problems. We first began by building our rectangle for the problem 8 x 6. My students build their rectangles with the vertical alignment of the first factor and then the horizontal alignment of the second factor (this helps tremendously down the road when we do division). Here’s a picture of it:

There were 8 of the 6 rods, so we built an 8 by 6 rectangle and we then needed to figure out the area inside the rectangle. I wanted to help students find an efficient way for figuring this out rather than just skip counting by six 8 times. So, I had them separate the rods like this:

That way, they could see that the 8 x 6 rectangle has the same area as two 4 x 6 rectangles. We discussed that we can think of any number times 8 as being the same product as a doubling of 4 times that number.

There were a couple of students in my small group who didn’t know that 4 x 6 was 24 so I had them separate their rods like this:

They could see that x4 is the same as x2 doubled. So, the 2×6 has an area of 12, we double that for 4×6 to equal 24 and then we double it again for 8 x 6 for a total of 48. After doing a few of these, the students were getting pretty good at figuring out some products with their x8 math facts. I then asked them if they could figure out what 8 x 16 would be. Within a couple of minutes they had the product of 128. (Down the road I know I’ll be discussing breaking apart the 16 by the place value positions, but this flexibility of thought is exactly what I’m trying to develop.) It was awesome! The students were so proud of themselves. There is no way anyone can memorize all the possible products, but by giving them a strategy for multiplying anything times 8, they were confident they could attempt it. Loved it!

A year and a half ago, I was offered the amazing opportunity to leave my 5th grade classroom of 13 years and become a Title 1 funded Math Coach K-5. I loved having my own classroom and teaching all subjects, but Math has always been my passion and the opportunity to share my passion with all teachers, assistants, parents, and students was simply a dream come true. I had a steep learning curve, though, since I knew nothing about how our youngest children learn math. So I read everything and anything on early numeracy and discovered the most amazing educators in cyberspace!! I was especially enamored and impressed with the educators in MTBoS. I have looked forward to checking my Twitter every day to see what new things I will learn, activities I can try in my classroom, and topics that really get me to think and reflect on my teaching practices. After seeing a particularly persuasive video of an MTBoS member encouraging more people to get involved, I decided that I wanted to become a more active member of this amazing group of educators and signed up for a mentor to help me get going. I couldn’t believe it when I learned that I was assigned Tracy Zager as my mentor since she was one of these amazing educators I was following on Twitter! Crazy cool!

My title “Setting the Foundation” serves a dual purpose with this blog post since it is my first post and, thus, is literally the foundation upon which I will build my blog, but it is also metaphorically a phrase that describes what has become my life’s work! The more that I work with students K-5, the more convinced I am that we need to teach them strategies that will help them succeed not only today but tomorrow as well. When we teach our first graders how to add two single-digit numbers by using strategies like make a ten, doubles plus 1, or bridge 10 to add over ten, we are not only helping them solve the problems they see today, but we are providing the foundation upon which they will progress to learn addition with larger numbers and even fractions and decimals down the line. When we teach 3rd graders how to build concrete models like an open array to show a multiplication problem, we are not only teaching them how to solve a problem today, but we are setting the foundation for using a model that will work with single and multi-digit whole numbers, then decimals in 5th grade, and even polynomials in algebra further on their math journey. It just make so much sense!

One last thing I want to mention in this first post…underlying all my work with teaching math facts strategically, encouraging the development of mental math strategies through Number Talks, and teaching models that will follow students as they progress on their journey, there is one foundational idea that I believe in my heart can change the course of anyone’s math journey – having a growth mindset. Jo Boaler’s ideas are revolutionary and so, so important. They encompass the idea that everyone can learn math! There is no “math brain”…our beliefs about math are shaped by our experiences and attitudes towards math. Our brains have the amazing ability to grow and change and by believing that we CAN learn by trying out new strategies and asking for help when we need it, we CAN learn things that we never dreamed we could learn! Helping children take chances and not be afraid of making mistakes…indeed, as Jo Boaler shares, brain research shows that when we are being challenged and we make a mistake trying out a new strategy, our brain grows! When we realize we made a mistake, it grows again, and when we fix it, it grows again. I want to be a part of this revolution that changes people’s mindsets about math and help those who have developed a phobia about math to see that math is beautiful and accessible to all! Thanks so much for joining me on this journey!